Colloquium: Atomic quantum gases in periodically driven optical lattices

Time periodic forcing in the form of coherent radiation is a standard tool\nfor the coherent manipulation of small quantum systems like single atoms. In\nthe last years, periodic driving has more and more also been considered as a\nmeans for the coherent control of many-body systems. In particular, experiments\nwith ultracold quantum gases in optical lattices subjected to periodic driving\nin the lower kilohertz regime have attracted a lot of attention. Milestones\ninclude the observation of dynamic localization, the dynamic control of the\nquantum phase transition between a bosonic superfluid and a Mott insulator, as\nwell as the dynamic creation of strong artificial magnetic fields and\ntopological band structures. This article reviews these recent experiments and\ntheir theoretical description. Moreover, fundamental properties of periodically\ndriven many-body systems are discussed within the framework of Floquet theory,\nincluding heating, relaxation dynamics, anomalous topological edge states, and\nthe response to slow parameter variations.\n